On linear Volterra equations in Banach spaces. (English) Zbl 0569.45020

The authors study the linear equation \(u'(t)=Au(t)+\int^{t}_{0}B(t- s)u(s)ds+f(t),\quad u(0)=u_ 0,\) in a Banach space X. They prove that there exists a reasonable resolvent operator if and only if the autonomous equation (where \(f=0)\) is well-posed (i.e., it has a unique solution that depends continuously on \(u_ 0)\). Furthermore, under some additional weak restrictions they show that a necessary and sufficient condition for this to happen is that \(| (1/n!)H^{(n)}(\lambda)| \leq M(Re \lambda -\omega)^{-n-1}\) for all Re \(\lambda\) \(>\omega\), \(n\geq 0\), where \(H(\lambda)=(\lambda -A-\hat B(\lambda))^{-1},\) that is a result of Hille-Yosida type. The authors also give an example showing that this condition can be satisfied although A does not generate a semigroup.
Reviewer: G.Gripenberg


45N05 Abstract integral equations, integral equations in abstract spaces
45D05 Volterra integral equations
Full Text: DOI


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